enigma123 wrote:
Ok - for GCD of 12 & 12
12 - 2,2,3
12- 2,2,3
So the common factors are 2 & 3. How come 12? Any guidance on GCD will be much appreciated. I say this with BIG please.
12= 2^2*3
24= 2^3*3
GCF(12, 24)= Product of minimum power of all common prime factors.
Locate the common prime factors; 2 and 3;
Let's check the minimum power of 2;
In 12: 2 has a power of 2.
In 24; 2 has a power of 3.
Here; 2<3
Thus; GCF will have 2^2(The minimum of the two powers)
Now;
In 12; 3 has a power of 1.
In 24; 3 has a power of 1.
Thus, minimum power of 3 is 1;
GCF will have 3^1
GCF=2^2*3^1=12
So; what's the GCF of 630 and 240.
630=3^2*5*2*7
240=2^4*5*3
Locate common prime factors; 2, 3 and 5.
Locate minimum powers of 2, 3 and 5 in both of these.
630 has 2 3's i.e. 3^2
240 has 1 3 i.e. 3^1
Thus, we consider: 3^1 for GCF
630 has 1 2 i.e. 2^1
240 has 4 2's i.e. 2^4
Thus, we consider 2^1 for GCF
630 has 1 5 i.e. 5^1
240 has 1 5 i.e. 5^1
Thus, we consider 5^1 for GCF
GCF(630,240)=3^1*2^1*5^1=30
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So, if we are given that GCF of z and 12 is 3, what do we know about z.
12=2^2*3
We know that z has at least one "3" in its factor AND z has no factor of 2 because even if there is one factor of 2 present in z, the GCF becomes 2^1*3^1=6, invalidating the statement; you see the point
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